Do POVM and generalized measurements really describe all possible measurements we can do on a quantum system (open dynamics)?

Question

My understanding in the motivation of POVMs and generalized measurements is that they allow to describe more general measurements than projective ones, if we don't describe the environment around the system we want to measure (if we allow to enlarge the system with an environment, and perform projective measurement on the bigger space, both are claimed to be equivalent).

In different terms, if I give you a density matrix $\rho_S$, and I forbid you to enlarge it with an environment, the set of generalized measurements is richer than the set of projective measurements you can do on this system.

Q1: do you agree so far?

Now, let's assume that I have access to $S$ and the environment $E$ around it. By doing projective measurements on $SE$ ($SE$ is closed), I can indirectly perform measurements on $S$ that couldn't be described by projective measurements on $S$ alone. This is where the notion of generalized measurements, and POVMs is useful.

I call $\\{ A_m \\}$ a family of operators describing a POVM, and I assume this POVM comes from a generalised measurement described by Kraus operators $\\{M_m\\}$ (thus I have $A_m=M_m^{\dagger} M_m$, this is always possible if I define $M_m=\sqrt{A_m}$).

The usual argument to say that every POVM can be seen as a projective measurement on a larger space consists to say that, for the initial state $\rho_S$, the post-measurement state would be:

$$\mathcal{E}(\rho)=\sum_m M_m \rho_S M_m^{\dagger}$$

$\mathcal{E}$ is a CPTP map thus it can be seen as the result of an entangling unitary between $S$ and $E$ (assuming they started in a product state) and a projective measurement done on $E$. I agree that it shows that every POVM can be interpreted as a projective measurement on an enlarged space. More precisely, as an entangling unitary applied on a product state on $SE$, followed by a projective measurement on $E$ alone.

However what I'm not sure to really understand is why every measurement we could conceptually do could be described by a local POVM/generalized measurement (on $S$).

For instance, I could imagine that $\rho_S$ is the reduced density matrix of an entangled state $|\psi_{SE}\rangle$ and I did a projective measuremement on $SE$ with projectors $\\{\Pi_k\\}$ (in general $\Pi_k$ can act non-trivially on the whole space $SE$). The reduced density matrix after the measurement, $\rho'_S$ would be equal to:

$$ \rho'_S= Tr_{E} \left( \sum_k \Pi_k |\psi_{SE}\rangle \langle \psi_{SE} | \Pi_k \right)$$ $$ \rho'_S= \sum_{k,u,v,v'} \langle u | \Pi_k | v \rangle \langle v | \psi_{SE}\rangle \langle \psi_{SE} | v' \rangle \langle v' | \Pi_k | u \rangle$$

In this last equality, the kets $| u \rangle,| v \rangle,| v' \rangle$ represent bases state of the environment.

I don't see how to define a family of local generalized measurement in this case (I'm not sure it is even possible: I think this is closely related to the fact CPTP do not represent the most general evolution on $S$ if $S$ and $E$ are initially entangled).

Q2: If it is not possible, wouldn't this mean that POVM and generalized measurement do not correspond to the most general measurements one can do (on a subsystem $S$ of an entangled system on $SE$)?

In different terms, there would exist measurements on $SE$ which will modify $\rho_S$ in a manner that cannot be described by a generalized measurement process. What disturbs me is that I think I have seen in some research papers some "security proof" for various protocol which assume an operator can do an arbitrary POVM (to show that the protocol is secure against any such attack). But wouldn't what I say here mean that checking security against every POVM is not sufficient? My last question is probably a bit blurry (I would need to find again such papers to make it clear, so you can ignore my last remark while answering if you don't see what I mean).

Answer 1

I would argue that your procedure does not correspond to a "measurement of system A" in an operational sense. For me, a measurement device should be able to take in a state $\rho_A$ (free to choose), and spit out a classical outcome with some probability (and maybe a post measurement state if you care about that). It should be able to do this regardless of which state we prepare on system A and how it is prepared.

However your process cannot take an arbitrary state, $\rho_A$ because the procedure depends explicitly on how $\rho_A$ was prepared. I.e. no manufacturer could prepare you such a device if they didnt also have control over your source. In particular, if the source changes then the measurement device also must change.

I would interpret your procedure as measuring the joint state $|\psi_{SE}\rangle$ rather than $\rho_A$.

With regards to your second question, all security proofs rely on assumptions, and most will have (implicitly) the assumption that the state of the measurement device is initially independent of the source. This assumption is arguably reasonable if you believe that you can freely choose which state to prepare and which measurement to perform independently of one another. Of couse you can find situations in which this is not reasonable, e.g. you bought your source and measurement device as one big box from a malicious manufacturer, then I see no reason why you can't consider a measurement scenario similar to what you describe in your question.

Answer 2

POVMs are the most general measurement as long as you insist that the probability $p_i(\rho)$ for any outcome $i$ is a linear function of $\rho$ (that is, any such linear $p_i(\rho)$ can be written as a POVM).

Answer 3

There's a few different things here, so let me try to break down a few different arguments:

1. Why are POVMs the most general possible measurement? — I'd say POVMs are simply what you get when applying the Born rule taking into account possible classical uncertainty. That is to say, if you can interpret any channel as a unitary dynamic in an enlarged space, and any density matrix as part of a larger pure state, and any measurement as a projective measurement in some basis of pure states with the usual Born rule for probabilities in the form $|\langle \psi|\phi\rangle|^2$, then you automatically get the formalism of POVM when you start neglecting ancillary degrees of freedoms. Vice versa, any POVM can be interpreted this way, eg via Naimark's theorem.

2. Post-measurement states — I don't agree that the channel $\mathcal E(\rho)=\sum_m M_m \rho_S M_m^\dagger$ describes "the post-measurement state" (or at least, it depends on what exactly you mean by that). This channel describes what you get on the system if you implement the POVM in the least destructive way possible — see e.g. Given a state $\rho$ and operator $0\le \Lambda\le I$, what does $\sqrt\Lambda \rho \sqrt\Lambda$ represent? —, and you neglect/don't know the measurement outcome. Otherwise, the post-measurement outcome conditional to observing the $m$-th outcome would be $M_m \rho_S M_m^\dagger/\operatorname{tr}(M_m \rho_S M_m^\dagger)$. Again, assuming a suitable way to implement the measurement that "minimally destroys" the measured state.

   If you want a channel that describes post-measurement states when you don't neglect the observed outcomes, you ought to use something like a quantum instrument instead.

3. Input-is-entangled-with-an-ancilla argument — The argument involving the initial system state being entangled with something else is a typical one also used when people try to argue that quantum channels are not the most general way to describe a quantum dynamic, and is (IMO) wrong for the same exact reason.

   One needs to be careful on what exactly is the problem you're trying to solve. If you are trying to describe measuring a state $\rho_S$, you can't say the the initial state is a purification $|\psi_{SE}\rangle$ of it. If the state you're measuring is $|\psi_{SE}\rangle$, then there's no problem with your expression, the Kraus operators would just be $M_{mk}= (I\otimes \langle m|)\Pi_k$ for some orthonormal basis for the ancillary space. But saying you're trying to measure/estimate something about $\rho$, implicitly means that you're ignoring or don't have access to the correlation between system and the purifying space. Which isn't the case if the state that's actually being measured is a purification of $\rho_S$.

   In fact, in such situation the problem is simply ill-defined. Suppose as a toy example that $S$ and $E$ are both single qubits, with $\rho_S=I/2$. Say the combined system $SE$ is measured with a (projective) Bell measurement. If the input is the purification $|\psi_{SE}\rangle=|00\rangle+|11\rangle$, then you'll always get the corresponding outcome when measuring. But if the purification is $|\psi_{SE}'\rangle=|01\rangle+|10\rangle$, you'll always get a different outcome. So it simply doesn't make sense to try to frame this as "measuring $\rho_S$", because if the measurement you're actually performing depends on the correlations/entanglement between $S$ and its purifying space, then the outcome probability distribution is a function of $|\psi_{SE}\rangle$, which cannot be cast as a function of $\rho_S$ alone.